Finite Group Databasebeta

← C22×#88942 #13181168

Subgroups — C22×#88942 #13181168

Order 1728.

1176 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1176 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6296C211yesmaximal, characteristic, complemented
2C22×#6295C211yesmaximal, characteristic
3C22×C3×#622C211yesmaximal, characteristic, complemented
4C2×#88942C2112yesmaximal, direct product factor
5C22×#3775—31nomaximal
6C23×C3×C3⋊D3—21no—
7C22×C3×C3⋊Dic3—21no—
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×#6296—23no—
10C2×#6296C2216yescomplemented
11C2×#6295—23no—
12C2×#6295C2216yes—
13C2×C3×#622C2216yescomplemented
14#88942C22116yesdirect product factor
15C22×#627—31no—
16C22×#627—61no—
17C22×#626—61no—
18C22×#625—31no—
19C22×#622D311yescharacteristic, complemented
20C2×#3775—312no—
21C23×C33D411yescharacteristic
22C22×C3×C3⋊D3—26no—
23C22×C3×C3⋊D3—44no—
24C2×C3×C3⋊Dic3—26no—
25C2×C3×C3⋊Dic3D413yes—
26C2×C3×C3⋊Dic3C2313yes—
27#6296—26no—
28#6296C2314yescomplemented
29#6295—26no—
30#6295C2314yes—
31C3×#622C2314yescomplemented
32C22×#204—91nomaximal
33C23×C3⋊D3—21no—
34C23×C3×D3—61no—
35C23×C3×D3—61no—
36C23×C3×D3—121no—
37C22×C3⋊Dic3—21no—
38C22×C3⋊Dic3C2×D311yescharacteristic
39C22×C3×Dic3—61no—
40C22×C3×Dic3—61no—
41C22×C3×Dic3—61no—
42C22×C3×Dic3—61no—
43C22×C3×Dic3—121no—
44C2×#627—36no—
45C2×#627—66no—
46C2×#627—66no—
47C2×#627—63no—
48C2×#626—66no—
49C2×#626—66no—
50C2×#625—36no—
51C2×#625—63no—
52C2×#622C2×D316yescomplemented
53#3775—316no—
54C22×C33SD1614yescomplemented
55C22×C33C2×D413yes—
56C2×C3×C3⋊D3—24no—
57C2×C3×C3⋊D3—412no—
58C3×C3⋊Dic3—24no—
59C3×C3⋊Dic3C2×D413yes—
60C3×C3⋊Dic3C2411yescharacteristic
61C22×#67—181no—
62C22×D12—91no—
63C22×Dic6—91no—
64C22×Dic6—181no—
65C22×C3×C8—91no—
66C2×#204—912no—
67C23×C32—21no—
68C23×C32—21no—
69C23×C32—41no—
70C23×C32—41no—
71C23×C32D1211yescharacteristic
72C22×C3⋊D3—26no—
73C22×C3⋊D3—44no—
74C22×C3×D3—66no—
75C22×C3×D3—66no—
76C22×C3×D3—126no—
77C22×C3×D3—124no—
78C22×C3×D3—128no—
79C22×C3×D3—124no—
80C2×C3⋊Dic3—26no—
81C2×C3⋊Dic3D1213yes—
82C2×C3⋊Dic3C22×D313yes—
83C2×C3×Dic3—63no—
84C2×C3×Dic3—63no—
85C2×C3×Dic3—63no—
86C2×C3×Dic3—63no—
87C2×C3×Dic3—66no—
88C2×C3×Dic3—66no—
89C2×C3×Dic3—126no—
90#627—34no—
91#627—64no—
92#627—612no—
93#627—66no—
94#626—64no—
95#626—612no—
96#625—34no—
97#625—66no—
98#622C22×D314yescomplemented
99C22×SD16—271noSylow
100C2×C33C2×SD1616yescomplemented
101C2×C33C22×D411yescharacteristic
102C3×C3⋊D3—44no—
103C24×C3—181no—
104C23×D3—61no—
105C23×D3—61no—
106C23×D3—121no—
107C22×C3×C4—91no—
108C22×C3×C4—181no—
109C2×#67—186no—
110C2×#67—186no—
111C22×Dic3—61no—
112C22×Dic3—61no—
113C22×Dic3—61no—
114C22×Dic3—61no—
115C22×Dic3—121no—
116C2×D12—96no—
117C2×D12—183no—
118C2×Dic6—96no—
119C2×Dic6—186no—
120C2×Dic6—183no—
121C2×Dic6—186no—
122C2×C3×C8—96no—
123#204—916no—
124C22×C32—23no—
125C22×C32—23no—
126C22×C32—24no—
127C22×C32—24no—
128C22×C32—44no—
129C22×C32—44no—
130C22×C32—43no—
131C22×C32—43no—
132C22×C32#20414yescomplemented
133C22×C32C2×D1213yes—
134C2×C3⋊D3—24no—
135C2×C3⋊D3—412no—
136C2×C3×D3—64no—
137C2×C3×D3—64no—
138C2×C3×D3—124no—
139C2×C3×D3—1212no—
140C2×C3×D3—1224no—
141C2×C3×D3—1212no—
142C3⋊Dic3—24no—
143C3⋊Dic3C2×D1213yes—
144C3⋊Dic3C23×D311yescharacteristic
145C3×Dic3—63no—
146C3×Dic3—63no—
147C3×Dic3—61no—
148C3×Dic3—61no—
149C3×Dic3—64no—
150C3×Dic3—64no—
151C3×Dic3—124no—
152C22×Q8—271no—
153C22×D4—271no—
154C2×SD16—2712no—
155C22×C8—91no—
156C33C22×SD1611yescharacteristic, complemented, Sylow
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—41no—
160C23×C3—41no—
161C23×C3—186no—
162C23×C3—364no—
163C23×C3GO+(4,𝔽2)11yescharacteristic
164C22×D3—66no—
165C22×D3—66no—
166C22×D3—126no—
167C22×D3—124no—
168C22×D3—128no—
169C22×D3—124no—
170C2×C3×C4—93no—
171C2×C3×C4—93no—
172C2×C3×C4—186no—
173#67—184no—
174#67—1812no—
175C2×Dic3—63no—
176C2×Dic3—63no—
177C2×Dic3—63no—
178C2×Dic3—63no—
179C2×Dic3—66no—
180C2×Dic3—66no—
181C2×Dic3—126no—
182D12—94no—
183D12—186no—
184Dic6—94no—
185Dic6—184no—
186Dic6—186no—
187Dic6—1812no—
188C3×C8—94no—
189C2×C32—21no—
190C2×C32—21no—
191C2×C32—26no—
192C2×C32—26no—
193C2×C32—46no—
194C2×C32—46no—
195C2×C32—41no—
196C2×C32—41no—
197C2×C32C2×#20416yescomplemented
198C2×C32C22×D1211yescharacteristic
199C3⋊D3—44no—
200C3×D3—124no—
201C3×D3—128no—
202C3×D3—124no—
203C24—181no—
204C2×Q8—276no—
205C2×Q8—543no—
206C2×D4—276no—
207C2×D4—543no—
208C22×C4—91no—
209C22×C4—181no—
210SD16—2716no—
211C2×C8—96no—
212C22×C3—23no—
213C22×C3—23no—
214C22×C3—24no—
215C22×C3—24no—
216C22×C3—43no—
217C22×C3—43no—
218C22×C3—44no—
219C22×C3—44no—
220C22×C3—184no—
221C22×C3—3612no—
222C22×C3#377514yescomplemented
223C22×C3C2×GO+(4,𝔽2)13yes—
224C2×D3—64no—
225C2×D3—64no—
226C2×D3—124no—
227C2×D3—1212no—
228C2×D3—1224no—
229C2×D3—1212no—
230C3×C4—91no—
231C3×C4—93no—
232C3×C4—184no—
233Dic3—63no—
234Dic3—61no—
235Dic3—63no—
236Dic3—61no—
237Dic3—64no—
238Dic3—64no—
239Dic3—124no—
240C32—21no—
241C32—21no—
242C32—41no—
243C32—41no—
244C32C22×#20411yescharacteristic, complemented
245C23—186no—
246C23—364no—
247C23#157⋊D311yescharacteristic
248Q8—274no—
249Q8—546no—
250D4—274no—
251D4—546no—
252C2×C4—93no—
253C2×C4—93no—
254C2×C4—186no—
255C8—94no—
256C2×C3—26no—
257C2×C3—26no—
258C2×C3—21no—
259C2×C3—21no—
260C2×C3—46no—
261C2×C3—46no—
262C2×C3—41no—
263C2×C3—41no—
264C2×C3—364no—
265C2×C3C2×#377516yescomplemented
266C2×C3C22×GO+(4,𝔽2)11yescharacteristic
267D3—124no—
268D3—128no—
269D3—124no—
270C22—184no—
271C22—3612no—
272C22#8894214yesdirect product factor
273C22C2×#157⋊D313yes—
274C4—91no—
275C4—93no—
276C4—184no—
277C3—21no—
278C3—21no—
279C3—41no—
280C3—41no—
281C3C22×#377511yescharacteristic, complemented
282C2—364no—
283C2C2×#8894216yesdirect product factor
284C2C22×#157⋊D311yescharacteristic
subgroupG/H
center, hypercenterC23#157⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×#157⋊D3
Fitting, socleC23×C33D4
prime radicalC22×#6295C2
hypocenterC33C22×SD16
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#6296C2semidirect
3. C22×C3×#622C2semidirect
4. C2×#88942C2direct (N × H)
10. C2×#6296C22semidirect
13. C2×C3×#622C22semidirect
14. #88942C22direct (N × H)
19. C22×#622D3semidirect
28. #6296C23semidirect
31. C3×#622C23semidirect
52. C2×#622C2×D3semidirect
54. C22×C33SD16semidirect
98. #622C22×D3semidirect
100. C2×C33C2×SD16semidirect
132. C22×C32#204semidirect
156. C33C22×SD16semidirect
197. C2×C32C2×#204semidirect
222. C22×C3#3775semidirect
244. C32C22×#204semidirect
265. C2×C3C2×#3775semidirect
272. C22#88942direct (N × H)
281. C3C22×#3775semidirect
283. C2C2×#88942direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#62962
C22×#62952
C22×C3×#6222
C2×#889422
C22×#37753
C22×#2049